Subgroups — #30774358

Order 17152.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569143	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22558299	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863953	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#16863526	—	2	1	no	—
8	#16863526	—	2	1	no	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	—	2	1	no	—
11	#16863422	C₄	1	2	yes	—
12	#16863420	C₄	1	1	yes	characteristic
13	#16863351	C₂²	1	2	yes	—
14	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#346×C₆₇	—	2	1	no	—
19	C₄×C₁₆×C₆₇	—	2	1	no	—
20	C₂×#11761612	C₂×C₄	1	2	yes	—
21	#14524043	C₈	1	1	yes	characteristic
22	#14524043	C₈	1	1	yes	characteristic
23	#14524035	C₈	1	2	yes	—
24	#14524022	—	2	1	no	—
25	#14524022	—	2	1	no	—
26	C₄×Dic₁₃₄	—	2	1	no	—
27	C₄×Dic₁₃₄	—	2	1	no	—
28	#14524002	C₂×C₄	1	2	yes	—
29	C₂×C₆₇⋊C₁₆	—	2	2	no	—
30	C₂×C₆₇⋊C₁₆	—	2	4	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	#14523953	—	2	2	no	—
36	C₈×Dic₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	2	2	no	—
41	M₃₂×C₆₇	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	—	2	1	no	—
49	#11761612	—	2	4	no	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₄⋊Dic₆₇	—	2	1	no	—
52	C₄⋊Dic₆₇	—	2	1	no	—
53	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
54	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	Dic₆₇⋊C₄	—	2	2	no	—
57	C₂²⋊Dic₆₇	—	2	2	no	—
58	C₄×Dic₆₇	—	2	2	no	—
59	C₆₇⋊C₁₆	—	4	2	no	—
60	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₆₇	#41	1	2	yes	—
70	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
74	C₁₆×C₆₇	—	2	4	no	—
75	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	4	1	no	—
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#131	1	2	yes	—
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	2	2	no	—
88	Dic₁₃₄	—	4	2	no	—
89	Dic₁₃₄	—	4	2	no	—
90	C₈×C₆₇	—	2	1	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₆₇	#124	1	1	yes	characteristic
94	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
95	C₆₇⋊C₈	—	4	2	no	—
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₆₇	#446	1	2	yes	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	#350	1	2	yes	—
104	C₄×C₆₇	#391	1	1	yes	characteristic
105	C₄×C₆₇	#408	1	1	yes	characteristic
106	C₄×C₆₇	C₈×Q₈	1	2	yes	—
107	Dic₆₇	—	4	2	no	—
108	Dic₆₇	—	4	2	no	—
109	#13265	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	#1984	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
113	C₂×C₆₇	#2117	1	1	yes	characteristic
114	#2960	—	67	1	no	—
115	C₂×C₄⋊C₁₆	—	67	1	no	—
116	C₄×M₃₂	D₆₇	1	1	yes	characteristic
117	#1379	—	67	4	no	—
118	C₆₇	#13265	1	1	yes	characteristic, complemented, Sylow
119	#533	—	67	1	no	—
120	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
121	C₂²×C₁₆	—	67	2	no	—
122	#446	—	134	1	no	—
123	#446	—	134	1	no	—
124	C₈×Q₈	—	134	1	no	—
125	C₈×Q₈	—	134	1	no	—
126	#424	—	67	2	no	—
127	#423	—	67	1	no	—
128	#405	—	67	2	no	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	C₄⋊C₁₆	—	134	1	no	—
131	C₄⋊C₁₆	—	134	1	no	—
132	#346	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₂×M₁₆	—	67	2	no	—
135	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	67	1	no	—
138	C₄⋊Q₈	—	67	1	no	—
139	#125	—	67	2	no	—
140	C₂²⋊Q₈	—	134	1	no	—
141	C₂²⋊Q₈	—	134	1	no	—
142	C₄×Q₈	—	134	1	no	—
143	C₄×Q₈	—	134	1	no	—
144	#117	—	67	2	no	—
145	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
146	M₃₂	—	2	2	no	—
147	M₃₂	Dic₁₃₄	1	4	yes	complemented
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	134	2	no	—
150	C₂×C₁₆	—	134	4	no	—
151	C₄⋊C₈	—	134	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₄⋊C₈	—	134	1	no	—
154	C₄⋊C₈	—	134	1	no	—
155	#97	—	134	2	no	—
156	C₄×C₈	—	134	2	no	—
157	C₄×C₈	D₂₆₈	1	1	yes	characteristic
158	C₄×C₈	D₂₆₈	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	134	1	no	—
162	C₂×Q₈	—	134	1	no	—
163	C₂²×C₄	#11761611	1	1	yes	characteristic
164	C₂²×C₄	#11761611	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
166	M₁₆	—	134	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	134	2	no	—
169	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
173	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
174	C₂×C₈	#11761628	1	2	yes	—
175	C₄⋊C₄	—	67	2	no	—
176	C₄⋊C₄	—	67	2	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	1	no	—
180	C₄⋊C₄	—	134	1	no	—
181	C₂²⋊C₄	—	134	2	no	—
182	C₄²	—	134	2	no	—
183	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
184	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₄²	C₈×D₆₇	1	1	yes	characteristic
186	C₄²	C₈×D₆₇	1	1	yes	characteristic
187	C₁₆	—	2	4	no	—
188	C₁₆	—	268	2	no	—
189	C₂³	C₂×#11761611	1	1	yes	characteristic
190	Q₈	—	268	2	no	—
191	Q₈	—	268	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	134	2	no	—
195	C₂×C₄	—	134	2	no	—
196	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
197	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
198	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
199	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
203	C₂×C₄	#14524049	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	268	2	no	—
207	C₈	#14524022	1	2	yes	—
208	C₈	#14524031	1	1	yes	characteristic
209	C₈	#14524043	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
213	C₂²	#16863526	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	268	2	no	—
218	C₄	—	268	2	no	—
219	C₄	#16863181	1	2	yes	—
220	C₄	#16863302	1	1	yes	characteristic
221	C₄	#16863362	1	1	yes	characteristic
222	C₄	C₈×Dic₁₃₄	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#22561145	1	1	yes	characteristic
225	C₂	C₂×#16863181	1	1	yes	characteristic
226	C₂	#22561838	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#13265
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
41. M₃₂×C₆₇	Q₈	semidirect
118. C₆₇	#13265	semidirect
147. M₃₂	Dic₁₃₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22569143	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558299	2
C₄×M₃₂×C₆₇	2
#13265	67